Size-121 non-right-cancellative idempotent magma built on the point set of the affine plane AG(2, 11), but with only 8 out of 12 parallel classes of lines preserved by the operation. These 88 'kept' lines are exactly the size-11 sub-quasigroups of the magma; pairs lying on a 'removed' line (slopes in F_11 P¹ where the line is NOT a sub-quasigroup) instead generate the whole magma under the closure of ◇.
Per-class α (in F_11 parameterization x ◇ y = (1-α)x + αy): 4×α=2 / 4×α=6.
• α = 2: parallel classes with F_11 slopes {0, 1, 3, 7}
• α = 6: parallel classes with F_11 slopes {4, 5, 10, INF}
Kept slopes (8): {0, 1, 3, 4, 5, 7, 10, INF}.
Removed slopes (4): {2, 6, 8, 9}.
Each column of the Cayley table is non-injective (collapse fibers), hence the magma is non-right-cancellative; nevertheless it is left-cancellative and every element is idempotent. Display reorder presents points as (h, v) ∈ F_11 × F_11; the 11 diagonal 11×11 blocks all render as the identical F_11(α=6) Cayley table.
[text written by Claude]
dwrensha · 2026-05-13 11:49:07